I Contradiction vs contraposition

Say I have the theorem ##p \rightarrow q##. What is the difference between proving that ##\neg q \rightarrow \neg p## is true and showing that ##\neg (p \rightarrow q) = \neg p \wedge q## leads to a contradiction?

- In proof by contraposition you start by assuming that [itex]\neg q[/itex] is true and derive the statement [itex]\neg p[/itex]. Here, the path is clear, i.e. you start at [itex]\neg q[/itex] and arrive at [itex]\neg p[/itex].

- In proof by contradiction your start by assuming that the opposite of [itex]p \rightarrow q[/itex] is true. So you assume that [itex]p \wedge \neg q[/itex] is true and derive some contradiction. Here the path is not clear, nobody is going to tell you what the contradiction is and what it looks like.